Generalized chaos expansions Lower bounds on Sobol indices · 2019. 10. 29. · Generalized chaos expansions Lower bounds on Sobol indices Olivier Roustant 1 ;2, Fabrice Gamboa 4,

Generalized chaos expansionsLower bounds on Sobol indices

Olivier Roustant 1,2, Fabrice Gamboa2,4, Bertrand Iooss2,3

1 INSA Toulouse2 Institut de Mathématiques de Toulouse

3 Electricité de France4 ANITI Artificial and Natural Intelligence Toulouse Institute

SAMO conference2019 October 28-30

O. Roustant, F. Gamboa, B. Iooss GCE, lower bounds on Sobol indices SAMO, 2019 October 28-30 1 / 24

Summary

1 With polynomial chaos, (total) Sobol indices are infinite sums of squaresI Sharp lower bounds are obtained by truncation (with equality cases)

2 The same is true for general tensors of orthonormal functionsI Generalized chaos expansion

3 When derivatives are available, we choose the orthonormal basis as theeigenfunctions of the Poincaré differential operator (PDO)

I PDO expansion


Part I

Context and notations



ANOVA decomposition

There is a unique decomposition for h ∈ L2(µ), with µ = µ1 ⊗ · · · ⊗ µd

h(x) = h0 +d∑

i=1

hi(xi) +∑i<j

hi,j(xi , xj) + · · ·+ h1,...,d (x1, . . . , xd )

s.t. E(hI(xI)|xJ) = 0 if J ( I. All the terms are orthogonal. Hence:

Var(h(x)) =d∑

i=1

Var(hi(xi)) +∑i<j

Var(hi,j(xi , xj)) + . . .

Sobol indicesPartial variances: DI = Var(hI(XI)), and Sobol indices SI = DI/D

D =∑

I

DI , 1 =∑

I

SI



Total indices and derivative-based measures

I ⊆ {1, . . . ,d} : DtotI =

∑J⊇I

DJ , νI =

∫ (∂|I|h(x)∂xI

)2

dµ(x)

These indices can be used to screen out useless variables or interactions:If either Dtot

i = 0 or νi = 0, then Xi is non influential


Derivative-based upper bounds of Sobol indices with Poincaréinequalities

Theorem [Lamboni et al., 2013]

The total Sobol index is bounded with DGSM, thanks to a Poincaré inequality:

Dtoti︸︷︷︸

interpretable but costly

≤ C(µi)︸︷︷︸Poincaré constant

×∫Rd

(∂h∂xi

(x))2

µ(dx)︸︷︷︸economical but less interpretable

Reminder: µ satisfies a Poincaré inequality if for all h in L2(µ) such that∫h(x)dµ(x) = 0, and h′(x) ∈ L2(µ):∫

h(x)2dµ(x) ≤ C(µ)

∫h′(x)2dµ(x)

→ Optimal values of C(µ) have been investigated in [Roustant et al., 2017]


Derivative-based upper bounds of Sobol indices with Poincaréinequalities

Actually, the same tool can be used to obtain lower bounds on Sobol indices!As for matricial problems, the minimum of the Rayleigh ratio (s.t.

∫hdµ = 0)∫

h′(x)2dµ(x)∫h(x)2dµ(x)

=‖h′‖2

‖h‖2

is given by the smallest eigenvalue of a spectral problem. More precisely, if µ1has density exp(−V ) on a bounded interval [a,b], then ∃(λn,en)n≥0 s.t. ∀h:

〈h′,e′n〉 = λn〈h,en〉 (?)

with 0 < λ1 = 1C(µ1)

< λ2 < · · · < λn → +∞.

Poincaré differential operator (PDO)

The underlying operator is Lh = h′′ − V ′h′, and solving (?) is equivalent to

Lh = −λh, with h′(a) = h′(b) = 0.

This can be solved numerically (fastly!) with 1-dimensional finite elements.


Part II

Generalized chaos expansion


Generalized chaos expansion

For all j , let ej,0 = 1,ej,1, . . . ,ej,nj−1 be orthonormal functions in L2(µj).We call generalized chaos a tensor of the form:

e`(x) =d∏

j=1

ej,`j (xj)

where ` = (`1, . . . , `d ) is a multi-index.When the e.,. are (orthonormal) polynomials, we recover polynomial chaos.

Property

The subset of tensors that involve exactly (resp. at least) x1 is an Hilbert basisof the corresponding space. Thus, for any centered function h:

h1 =∑`1≥1

〈h,e1,`1〉e1,`1 , htot1 =

∑`1≥1,`2,...,`d

〈h,e`〉e`

D1(h) =∑`1≥1

〈h,e1,`1〉2, Dtot1 (h) =

∑`1≥1,`2,...,`d

〈h,e`〉2

N.B. This material is inspired from [Antoniadis, 1984, Tissot, 2012].O. Roustant, F. Gamboa, B. Iooss GCE, lower bounds on Sobol indices SAMO, 2019 October 28-30 9 / 24

PDO expansions

Define PDO expansion as the generalized chaos expansion obtained withthe eigenfunctions of the Poincaré differential operator.

Property

D1(h) =∑`1≥1

〈h,e1,`1〉2 =∑`1≥1

1λ2

1,`1

〈 ∂h∂x1

,e′1,`1〉2.

Dtot1 (h) =

∑`1≥1,`2,...,`d

〈h,e1,`1 . . . ed,`d 〉2

=∑

`1≥1,`2,...,`d

1λ2

1,`1

〈 ∂h∂x1

,e′1,`1e2,`2 . . . ed,`d 〉2.


PDO expansions

Example of lower bound. Limiting ourselves to the first eigenfunction in alldimensions, and to first and second order tensors involving x1, we obtain:

A derivative-free PDO lower bound:

Dtot1 (h) ≥ 〈h,e1,1〉2︸︷︷︸

lower bound for D1

+d∑

i=2

〈h,e1,1ei,1〉2

A derivative-based PDO lower bound:

Dtot1 (h) ≥ C(µ1)

2 〈 ∂h∂x1

,e′1,1〉2︸︷︷︸lower bound for D1

+ C(µ1)2

d∑i=2

〈 ∂h∂x1

,e′1,1ei,1〉2

Equality case: when h has the form

h(x) = α1e1,1(x1) +∑d

i=2 αie1,1(x1)ei,1(xi) + g(x2, . . . , xd )


Particular cases

For uniform distributions, PDO expansion = Fourier expansionI Indeed, the PDO is the Laplacian, whose eigenfunctions are trigo. functions

For normal distributions, PDO expansion = PC expansionI This is the only case where PDO expansion = PC expansion


Improvements on existing works (in [Kucherenko and Iooss, 2017])

For uniforms on [0,1] using the orthonormal function obtained from xm1 ,

and an integration by part, we obtain:

Dtot1 ≥ D1 ≥

2m + 1m2

(∫(g(1, x−1)− g(x))dx − w (m+1)

1

)2

where w (m+1)1 =

∫ ∂g(x)∂x1

xm+11 dx . This improves on the known lower bound

which has the same form, with the smaller multiplicative constant 2m+1(m+1)2 .

For normal distributions, we improve on:

Dtot1 ≥ D1 ≥ s2

1

(∫∂g(x)∂x1

dµ(x))2

.

N.B. Better bounds are obtained by adding orth. funct. of the form ψ1ψj .


Part III

An application


A case study for global sensitivity analysis

A simplified flood model [Iooss, 2011], [Iooss and Lemaitre, 2015].Output: cost (in million euros) of the damage on the dyke

Y = 1IS>0 +[0.2 + 0.8

(1− exp−

1000S4

)]1IS≤0 +

120

(Hd1IHd>8 + 81IHd≤8)

where H is the maximal annual height of the river (in meters), and S isthe maximal annual overflow (in meters)

S = Zv + H − Hd − Cb with H =

Q

BKs

√Zm−Zv

L

0.6


A case study for global sensitivity analysis

8 inputs variables assumed to be independent r.v., with distributions:

Input Description Unit Probability distributionX1 = Q Maximal annual flowrate m3/s Gumbel G(1013, 558),

truncated on [500, 3000]X2 = Ks Strickler coefficient - Normal N (30, 82),

truncated on [15,+∞[X3 = Zv River downstream level m Triangular T (49, 50, 51)X4 = Zm River upstream level m Triangular T (54, 55, 56)X5 = Hd Dyke height m Uniform U [7, 9]X6 = Cb Bank level m Triangular T (55, 55.5, 56)X7 = L River stretch m Triangular T (4990, 5000, 5010)X8 = B River width m Triangular T (295, 300, 305)


Illustration on the flood problem: PDO lower bounds without derivatives

100 1000 10000 1e+05

0.0

0.2

0.4

0.6

0.8

Q0.

00.

20.

40.

60.

8

100 1000 10000 1e+05

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Ks

0.0

0.1

0.2

0.3

0.4

0.5

0.6

100 1000 10000 1e+05

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Zv

0.0

0.1

0.2

0.3

0.4

0.5

0.6

100 1000 10000 1e+05

0.00

0.05

0.10

0.15

0.20

Zm

0.00

0.05

0.10

0.15

0.20

100 1000 10000 1e+05

0.00

0.05

0.10

0.15

0.20

Hd

0.00

0.05

0.10

0.15

0.20

100 1000 10000 1e+05

0.00

0.05

0.10

0.15

0.20

Cb

0.00

0.05

0.10

0.15

0.20

100 1000 10000 1e+05

0.00

0.05

0.10

0.15

0.20

L

0.00

0.05

0.10

0.15

0.20

100 1000 10000 1e+05

0.00

0.05

0.10

0.15

0.20

B

0.00

0.05

0.10

0.15

0.20

MC estimate of PDO lower bound of (total) Sobol indices, for various sample sizes:

Dtot1 ≥ 〈h, e1,1〉2︸︷︷︸

lower bound for D1

+∑d

i=2〈h, e1,1ei,1〉2


Illustration on the flood problem: PDO lower bounds using derivatives

100 1000 10000 1e+05

0.0

0.2

0.4

0.6

0.8

Q0.

00.

20.

40.

60.

8

100 1000 10000 1e+05

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Ks

0.0

0.1

0.2

0.3

0.4

0.5

0.6

100 1000 10000 1e+05

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Zv

0.0

0.1

0.2

0.3

0.4

0.5

0.6

100 1000 10000 1e+05

0.00

0.05

0.10

0.15

0.20

Zm

0.00

0.05

0.10

0.15

0.20

100 1000 10000 1e+05

0.00

0.05

0.10

0.15

0.20

Hd

0.00

0.05

0.10

0.15

0.20

100 1000 10000 1e+05

0.00

0.05

0.10

0.15

0.20

Cb0.

000.

050.

100.

150.

20

100 1000 10000 1e+05

0.00

0.05

0.10

0.15

0.20

L

0.00

0.05

0.10

0.15

0.20

100 1000 10000 1e+05

0.00

0.05

0.10

0.15

0.20

B

0.00

0.05

0.10

0.15

0.20

MC estimate of PDO lower bound of (total) Sobol indices, for various sample sizes:

Dtot1 ≥ C(µ1)

2 〈 ∂h∂x1

, e′1,1〉2︸︷︷︸

lower bound for D1

+ C(µ1)2

d∑i=2

〈 ∂h∂x1

, e′1,1ei,1〉2


Conclusions on the application

1 Lower bounds are easily computed, even for exotic input distributions

2 The estimation error can be large for small sample sizesI Bootstrap confidence intervals are required

3 The (estimated) lower bounds of the total Sobol’ indices are ofteninformative, i.e. larger than the (estimated) first order Sobol’ indices

4 Using derivatives (then DGSM) gives excellent results, even for smallsample size cases


Part IV

Conclusions and perspectives


Take-home messages

1 Polynomial chaos (PC) expansion is extended to tensor Hilbert basesI Gives lower bound for Sobol indices, with equality cases

2 When derivatives are available, a good Hilbert basis is given by theeigenfunctions of the Poincaré Differential Operator (PDO expansion)

I Suitable lower bounds for Sobol indices are obtained with first eigenvaluesI Improves on existing results on derivative-based sensitivity measures

3 PDO expansion can be computed fastly for various prob. distributionsI 1-dimensional finite element methods

4 PDO expansion 6= PC expansion, except for the Normal distributionI Only two other exceptions, when using weights: Gamma & Beta.I For the uniform distribution, PDO expansion = Fourier expansion.


Perspectives

1 To investigate finite sample properties of estimatorsI Reduce bias for small sample size in both PDO and PC expansions

2 To adapt L1 techniques for PDO expansionsI In order to choose relevant terms (not only the first eigenvalues)

3 To compare PDO and PC expansions in engineering problems

To go further into details, discover the preprint here


https://hal.archives-ouvertes.fr/hal-02140127

Acknowledgements

Aldéric Joulin, for fruitful discussions on differential operatorsOQUAIDO ChairANR-3IA Artificial and Natural Intelligence Toulouse Institute

Thank you for your attention!


https://oquaido.emse.fr

https://en.univ-toulouse.fr/aniti

References

Antoniadis, A. (1984).Analysis of variance on function spaces.Statistics: A Journal of Theoretical and Applied Statistics, 15(1):59–71.

Bakry, D., Gentil, I., and Ledoux, M. (2014).Analysis and geometry of Markov diffusion operators, volume 348 of Grundlehren der MathematischenWissenschaften [Fundamental Principles of Mathematical Sciences].Springer, Cham.

Iooss, B. (2011).Revue sur l’analyse de sensibilité globale de modèles numériques.Journal de la Société Francaise de Statistique, 152:1–23.

Iooss, B. and Lemaitre, P. (2015).A review on global sensitivity analysis methods.In Meloni, C. and Dellino, G., editors, Uncertainty management in Simulation-Optimization of ComplexSystems: Algorithms and Applications, pages 101–122. Springer.

Kucherenko, S. and Iooss, B. (2017).Derivative-based global sensitivity measures.In Ghanem, R., Higdon, D., and Owhadi, H., editors, Springer Handbook on Uncertainty Quantification,pages 1241–1263. Springer.

Lamboni, M., Iooss, B., Popelin, A.-L., and Gamboa, F. (2013).Derivative-based global sensitivity measures: General links with Sobol’ indices and numerical tests.Mathematics and Computers in Simulation, 87:45–54.

Roustant, O., Barthe, F., and Iooss, B. (2017).Poincaré inequalities on intervals - application to sensitivity analysis.Electron. J. Statist., 11(2):3081–3119.

Song, S., Zhou, T., Wang, L., Kucherenko, S., and Lu, Z. (2019).


References

Derivative-based new upper bound of Sobol’ sensitivity measure.Reliability Engineering & System Safety, 187:142 – 148.

Tissot, J.-Y. (2012).Sur la décomposition ANOVA et l’estimation des indices de Sobol’. Application à un modèled’écosystème marin.PhD thesis, Grenoble University.


References

Appendix. Weights and connexion with PC expansions

1 PDO expansion can be extended to weighted Poincaré inequalities,

Varµ1(h) ≤ C∫R

h′(x)2w(x)µ1(dx)

by solving 〈h′,e′n〉w = λn〈h,en〉 with 〈h,g〉w :=∫

h(x)g(x)w(x)µ(dx).

2 Using weighted Poincaré inequalities has already been proposed in SA:[Song et al., 2019] choose w such that e1 is a 1st order polynomial.→ Except from 3 cases, the other eigenfunctions en are not all polynomials.

3 There are exactly 3 cases where PDO expansions = PC expansions,i.e. where all eigenfunctions are polynomials [Bakry et al., 2014]:

Law Interval Polynomials WeightNormal R Hermite w(x) = xGamma R+ Laguerre w(x) ∝ xα−1e−αx

Beta [−1,1] Jacobi w(x) ∝ (1− x)α−1(1 + x)β−1


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Generalized chaos expansions Lower bounds on Sobol indices · 2019. 10. 29. · Generalized chaos expansions Lower bounds on Sobol indices Olivier Roustant 1 ;2, Fabrice Gamboa 4,